On the Sylow graph of a group and Sylow normalizers

الأصل · EN

Let G be a finite group and Gₚ be a Sylow p-subgroup of G for a prime p in π(G), the set of all prime divisors of the order of G. The automiser Aₚ(G) is defined to be the group NG(Gₚ)/GₚCG(Gₚ). We define the Sylow graph Γₐ(G) of the group G, with set of vertices π(G), as follows: Two vertices p,q∈π(G) form an edge of Γₐ(G) if either q∈π(Aₚ(G)) or p∈ π(Aq(G)). The following result is obtained: Theorem: Let G be a finite almost simple group. Then the graph Γₐ(G) is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.